Related papers: Fault-Equivalent Lowest Common Ancestors

We study minimum vertex cover problems on random \alpha-uniform hypergraphs using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase…

We study minimal vertex covers of trees. Contrarily to the number $N_{vc}(A)$ of minimal vertex covers of the tree $A$, $\log N_{vc}(A)$ is a self-averaging quantity. We show that, for large sizes $n$, $\lim_{n\to +\infty} <\log…

We study whether iterated vector fields (vector fields composed with themselves) are conservative. We give explicit examples of vector fields for which this self-composition preserves conservatism. Notably, this includes gradient vector…

Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive…

We study the fault-tolerance of networks from both the structural and computational point of view using the minimum leaf number of the corresponding graph $G$, i.e. the minimum number of leaves of the spanning trees of $G$, and its…

The 2-Decomposition Conjecture, equivalent to the 3-Decomposition Conjecture stated in 2011 by Hoffmann-Ostenhof, claims that every connected graph $G$ with vertices of degree 2 and 3, for which $G \setminus E(C)$ is disconnected for every…

Federated Learning (FL) is an emerging paradigm that allows a model to be trained across a number of participants without sharing data. Recent works have begun to consider the effects of using pre-trained models as an initialization point…

We investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence. We show that such trees are caterpillars and that the vertex degrees are non-decreasing on every path on…

Balanced minimum evolution is a distance-based criterion for the reconstruction of phylogenetic trees. Several algorithms exist to find the optimal tree with respect to this criterion. One approach is to minimize a certain linear functional…

Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families…

Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each…

The resiliency of a network is its ability to remain \emph{effectively} functioning also when any of its nodes or links fails. However, to reduce operational and set-up costs, a network should be small in size, and this conflicts with the…

Let $X \subset {\mathbb R}^m$ be a spherical code (i.e., a finite subset of the unit sphere) and consider the ideal of all polynomials in $m$ variables which vanish on $X$. Motivated by a study of cometric ($Q$-polynomial) association…

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1…

Let $[n] = \{1, 2, \ldots, n\}$ and let $2^{[n]}$ be the collection of all subsets of $[n]$ ordered by inclusion. ${\cal C} \subseteq 2^{[n]}$ is a {\em cutset} if it meets every maximal chain in $2^{[n]}$, and the {\em width} of ${\cal C}…

Low-Rank Adaptation (LoRA) is a popular technique for efficient fine-tuning of foundation models. However, applying LoRA in federated learning environments, where data is distributed across multiple clients, presents unique challenges.…

We study a pathfinding problem where only locations (i.e., vertices) are given, and edges are implicitly defined by an oracle answering the connectivity of two locations. Despite its simple structure, this problem becomes non-trivial with a…

For a connected graph, a vertex separator is a set of vertices whose removal creates at least two components and a minimum vertex separator is a vertex separator of least cardinality. The vertex connectivity refers to the size of a minimum…

We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$…

Limited training data and severe class imbalance impose significant challenges to developing clinically robust deep learning models. Federated learning (FL) addresses the former by enabling different medical clients to collaboratively train…